Approximately transitive flows and ITPFI factors

Ergod. Th. & Dynam. Sys. (1985), 5, 203-236Printed in Great Britain

Approximately transitive flows andITPFI factors

A. CONNES AND E. J. WOODS

InstitutedesHautes EtudesScientifiques, 35, routede Chartres, 91440 Bures-sur-Yvette,France; Department of Mathematics and Statistics, Queen's University, Kingston,

Ontario, K1L 37V6, Canada

{Received 27 October 1983)

This paper is dedicated to Richard V. Kadison on the occasion of his sixtieth birthday.

Abstract. We define a new property of a Borel group action on a Lebesgue measurespace, which we call approximate transitivity. Our main results are (i) a type III0

hyperfinite factor is ITPFI if and only if its flow of weights is approximately transitive,and (ii) for ergodic transformations preserving a finite measure, approximate transi-tivity implies zero entropy.

0. Introductionvon Neumann algebras are the non-commutative analogue of measure theory spaces.The product measures of measures on finite sets give rise to a class of factors calledITPFI factors (see terminology). However, in the classification problem the mostnatural class turned out to be the approximately type I factors [4] (those factorswhich are well-approximated by finite-dimensional ones - see terminology). It istrivial that ITPFI implies approximately type I, but the converse is false andnon-trivial [12], [2], [6]. (Since all non-atomic standard Borel measures are Borelisomorphic, the corresponding problem does not arise in measure theory.) TheITPFI factors are certainly the most natural subclass of the approximately type Ifactors. Their exact position among the approximately type I factors (up to isomorph-ism of factors) has remained an interesting mystery for some time. In particular,there is still no direct spatial construction of a non-ITPFI approximately type Ifactor. The crucial existential step is always carried out in the flow of weights (anergodic flow which is naturally defined as an invariant of the factor). The problemonly arises for factors of type III0 or IIIi. The known examples of non-ITPFIapproximately type I factors are all of type III0. In this paper we completelycharacterize the ITPFI factors among the type III0 approximately type I factors bya new ergodic property of their complete invariant, the flow of weights, which we

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204 A. Connes and E. J. Woods

call approximate transitivity. Of course this transfers the original problem to under-standing approximate transitivity for ergodic flows. Our second major result is thatfor finite measure preserving flows, approximate transitivity implies zero entropy.

Approximate transitivity is a new and apparently interesting notion in ergodictheory. Our proof that ITPFI implies approximate transitivity is rather straightfor-ward (lemma 8.1) and is, in fact, how this property was discovered. Our proof ofthe converse is of particular interest because it is obtained by attempting directlyto 'invert the flow of weights arrow'. The argument is quite similar to the Murray-vonNeumann proof of the uniqueness of the hyperfinite IIt factor [15]. They embed afinite-dimensional algebra in a finite type I factor in a very precise manner relativeto the trace. For type III factors no trace exists and, instead of only comparingminimal projections inside the finite-dimensional algebra, we compare such minimalprojections together with the restriction of a state to these projections. This com-parison directly yields measure theoretical objects on the flow of weights, and (exceptin the III, case) is non-trivial even though the comparison of projections is trivial.Our paper is intended to illustrate this technique. In fact an alternate proof ofKrieger's theorem, not using the cohomological technique of Krieger, can be basedon the original Murray-von Neumann proof of the uniqueness of the hyperfiniteII, together with the above refined comparison of projections. (Krieger's theorem[13] states, in part, that the flow of weights considered as a mapping from type III0

Krieger factors with algebraic isomorphism as the equivalence relation, to strictlyergodic flows with conjugacy as the equivalence relation, is one-to-one and ontobetween equivalence classes.)

It is easy to translate our proof to the purely ergodic setting of non-singulartransformations. Our result would then follow from Krieger's theorem (our proofdoes not use Krieger's theorem). However, as mentioned above, part of our goalwas to exhibit the flow of weights as a useful technique. Indeed we present moreof the comparison theory of finite weights than is needed for our proof.

§ 1 contains some terminology. In § 2 we define approximate transitivity (hereafterreferred to as AT) for Borel group actions and give some elementary properties. In§ 3 we prove that for finite measure preserving transformations, AT implies zeroentropy. In § 4 we give three different constructions of the flow of weights whichwill be used later. § 5 contains a comparison theory for finite periodic weights, and§ 6 gives the comparison theory for finite (not necessarily periodic) weights. In § 7we introduce a 'product property' which is equivalent to being ITPFI. In § 8 weprove the equivalence of the ITPFI and AT properties.

1. TerminologyA von Neumann algebra M is said to be approximately type I if it is of the form

M = UMn ,\n-\ I

where Mn <= Mn+I for each n, and each Mn is a finite-dimensional matrix algebra(the names approximately finite, hyperfinite, approximately finite dimensional, and

Approximately transitive flows 205

matricial have all been used in the literature for this concept). A factor M is saidto be ITPFI if it is of the form M = (§)™=1 {Mn, <f>n) where each Mn is a finite typeI factor ([1], [21]). A factor M is called a Krieger factor if it can be obtained froman ergodic action of Z by the Murray-von Neumann group measure space construc-tion. (It is straightforward that ITPFI =>Krieger=>approximately type I.)

For a detailed explanation of the following standard terminology see, for example[20]. The term weight always means a normal semi-finite weight. If 0 is a weighton the von Neumann algebra M then o-f is the modular automorphism group. Theinvariant T(M) is the set of all t such that erf is inner. M^ denotes the predual ofM, and M% is then the set of all finite weights. If <Ae Af* then s(i/0 denotes thesupport of if/. The flow of weights of M is an ergodic action of Rj on some measurespace (XM, fj.M). The construction of [5] gives not that measure space, but themeasure algebra whose elements are equivalence classes [] of integrable weights$ of infinite multiplicity. The flow is then defined by f*[] = [t<t>]. It is sometimesconvenient to consider the flow as an action of U, in which case it is written F™ = S1™(if M is understood, it is usually omitted).

If / is a function on a measure space (X, fi) then ||/ | | denotes the L'-norm of/If fi, v are finite measures on X then ||/A - »>|| is the L'-norm defined by \\dfi/da-dv/da\\ where fi,p<a. If x is a finite weight or operator then ||x|| denotes theusual norm.

2. AT actions-elementary propertiesWe define approximate transitivity of a Borel group action on a Lebesgue measurespace, and establish some elementary properties.

Definition 2.1. Let G be a Borel group, (X, v) a Lebesgue measure space, anda: G-» Aut (X, v) a Borel homomorphism. We say that the action is approximatelytransitive (AT) if given n < oo, finite measures fiit..., /*.„ < v, and e > 0, there existsa finite measure /u. < v, gu... ,gmeG for some m < oo, and Ajt>0, k = 1 , . . . , msuch that

- I •e, j = ! , . . . , « . (2.1)

If G = Z and a is AT, then we say that T= a ( l ) is AT.

Remark 2.2. There are a number of elementary variations on this definition.(i) The index k need not be restricted to a finite set. Typically we will take keZ

and consider \Jk as a function Aj-e/+(Z).(ii) One can demand that ||^|| = 1 and ||A,|| = £fc \jk = \\fij\\.(iii) By taking \\fi\\ sufficiently small, one can take the \Jk to be integers.(iv) It is sufficient to ask that eq. (2.1) hold for n=2. (If /*' approximates

/* i , . . . , )ttn_, in the sense of eq. (2.1), choose fi to approximate fi' and /*„)•(v) For continuous actions of a locally compact group, the kjk can be replaced

by functions A7 e L\(G, dg) such that

•e, j=l,...,n. (2.2)

To show that eq. (2.2) implies eq. (2.1), approximate the A;(g) by simple functions.To prove the converse, write the sum as an integral over delta functions 5(ggfc')A k

and then approximate by functions in L\.(G, dg).(vi) The equation h = dcr/di> gives a one-to-one correspondence between func-

tions h e L+(X, v) and finite measures a <v. We have

d(aga-)/dv = pgag(da/ dv),

where pg = d(agv)/dv. Then

(Ptf)(x)=f(g-lx)Pg(x) (2.3)

defines a homomorphism /3 from G into the invertible isometries on L[(X, v). It istherefore equivalent to ask that for any / , , . . . , / „ e L\(X, v) and e > 0, there exist/ e L\(X, v), g , , . . . , gm e G and \Jk > 0 such that

It m IIp " ^ / ^ ' ^ " e » 7 = 1 , •••,«• (2.4)

If Og = i' for all geG, then fi = a.

LEMMA 2.3. An AT action is ergodic.

Proof. Let BcX,v(B)> 0, i / (X\B) > 0, and agB = B for all geG. Choose B, c Band B 2 c X \ B such that 0(B,-)<<», 7 = 1,2. Let Hj=v\Bj. Then eq. (2.1) for7 = 2 implies that fi(B)<e. If e<5^(B,) this contradicts eq. (2.1) forj = 1. D

Remark 2.4. Let (X, SB, i', G, a) be a Borel group action, *>(X)<oo. Let S80 be asub-o--algebra of the <r-algebra 38 of Borel subsets of X, such that ag3ft0= S80 forall geG. Then the restriction (X, @l0, v,G,a) is called a. factor action of the givenaction. If a- is any finite measure on (X, 38) we have Hfllcx.aso)— Il°"ll(x,s8)- Henceany factor action of an AT action is again AT.

The base and ceiling function construction of a flow is particularly useful when theceiling function is constant. In this situation one naturally expects that the flow willhave a certain property if and only if the base transformation has the correspondingproperty.

LEMMA 2.5. Let (X, v, Fs) be a flow built over the base transformation (B, vB, T)with a ceiling function with constant height H. Then Fs is AT if and only if T is AT.

Proof. We can write X = BxI where the interval / = [0, H) carries Lebesguemeasure. We have

Fs(b,t) = (T"b,u), (2.5)

where s +1 = nH + u, n e Z , 0 < w < HAssume that T is AT. Let /u.u..., fin< v, IXJ(X) < 00, and e > 0. Since rectangles

generate the measure algebra, it follows by a straighforward but tedious argumentthat one can approximate the /t,- by a sum of product measures. More precisely,there exists an integer L and measures fiJk, k = 0 , . . . , L - 1 on B such that

fl~ I If/* x mk\\ <e, j=l,...,n (2.6)

where mk is the restriction of Lebesgue measure to the interval Lk =[kH/L, (k+l)H/L). Since T is AT there exist crB< vB and \jke /+(Z) such that

Define

and

Then

- IqeZ

cr = crB X m0

\j(qL+ k) = kjk{q),

<eL~ (2.7)

(2.8)

(2.9)

- I *j(r)FrH/L<r <e •V ( W x e * - I Xj(r)FrH/Lo)k = 0 I \ reZ /

L - l

<2e. (2.10)

Thus F, is AT.Now assume that Fs is AT. Let fit,..., ixn< vB, /x,(B)=l, and 0 < e < | .

Let Jk = [kH/6, (k+\)H/6), k = 0, 1. . . . .5, and let mk denote the restriction ofLebesgue measure to Jk. Let

iij = HXm3, j=\,...,n. (2.11)

Then there exist /x < v, ||/i|| = 1 and Aj€<f+(Z), ||A || = 1, and skeU, fceZ such that

Xj(k)F,,ji

In particular we have

<e/6, 7 = 1 , . . . , » .

<e /6 .

(2.12)

(2.13)

In order to produce the desired measure a- on B and Aj€^+(Z) it is necessary torestrict the supports of Ay and //. somewhat. Since ||/*|| = 1 there is some 0< K < 5such that

MBXJJH. (2.14)

By shifting the sk and shifting /i under Fs (if necessary) we can assume that K — 4.Let

Then fc^ y implies that (FSk(^i\BxjA))(BxJ3) = 0. It now follows from eqs. (2.13)and (2.14) that

(2.16)

and thus

(2.17)

Write /x = fi' + /A" where

/*"= I

If fee Y then (FSkv")(B x73) = 0, and eqs. (2.13) and (2.17) imply that

Eqs. (2.17) and (2.19) imply that

\j(k)FSk,x- I Xj(k)FsyIceZ

(2.18)

(2.19)

(2.20)

Let P%cr denote the canonical projection of the finite measure cr on X = B x / ontoB. We have

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

The tower construction of a single transformation (see for example [9]) is theanalogue of the base and ceiling function construction of a flow.

COROLLARY 2.6. Let the transformation (X, v, S) be constructed as a tower over(B, vB, T) with constant height H. Then S is AT if and only if T is AT.

Proof. The flow Fs built over (X, v, S) with constant height one is obviously aflow built over (B, vB, T) with constant height H. The result now follows fromlemma 2.5. •

Recall that a finite measure preserving transformation is said to have rank one ifthere is a sequence of Rohlin towers which approximate the measure algebra. Moreprecisely one asks that given/,, . . . , / „ £ L\x, v) and e > 0, there exist B c X, m < oo,and \JkeIR such that B,TB,..., TmB are disjoint and

and

if sk e yp where

(It is eq. (2.22) that depends crucially on the support properties.) Let

A,(p)= I A,(fc).

Since P* is norm decreasing, eqs. (2.12), (2.20) and (2.21)-(2.24) give

M,- I Aj(p)Tpo- < 8 H - ' e ,

where cr =

k=07 = 1 , . . . , n. (2.26)

(Note that if j J s O we can require that XJk>0.) A slightly weaker condition, calledfunny rank one, is obtained by replacing the sequence B, TB,..., TmB by the setsT"°B, T"'B,..., T"mB where {«,} is an arbitrary sequence (depending on / , , . . . ,/„and e}.

LEMMA 2.7. Given (X, T, v) with Tv — v, v(X) = 1. Then funny rank one implies AT.

Proof. It is convenient here to use the AT condition on functions in L+(X, v) (seeremark 2.2 (iv)). L e t / , . . . , / n eL+(X, v), e > 0 . Then there exist Be X, a sequence{nk}k=0 m and \jk such that

U - Z ijkXr-kB >e. (2.27)II k = 1

Let

A* = (A-'t ' (2-28)

10 i/A; k<0.

Since the T"kB are disjoint, we have

II f -Y A'tT""*fll< II/"--y A tT~"'tf!l (2 29111 / / Z* A j l c l J II — l l / j Zw / vj(c J 7 ll» V z - Z " /

where / = * B - DCOROLLARY 2.8. / / (X, T, *>) fcas pure point spectrum then T is AT.Proof. Pure point spectrum implies rank one [11]. •

COROLLARY 2.9. A pure point spectrum flow is AT.

Proof. Such a flow can be built over a pure point base transformation with a constantceiling function. The result now folllows from corollary 2.8 and lemma 2.5. D

It is also known that certain diffeomorphisms of the circle are AT [10].

3. AT transformations and entropyIn this section we prove that if (X, /u,, T) is a finite measure preserving transformation,then AT implies that T has zero entropy (theorem 3.5).

In analyzing the implications of the AT condition one immediately observes thatan expression of the form

T(\)f = y^\jTif (3.1)

where \ef+(Z) and / e L+(X, /x), is in effect a convolution, and 'convolutionsspread functions out'. This spreading can present some difficulties when one triesto satisfy the AT condition for functions/ =XA1,fi = XA2 where Au A2<= X. In orderto give a precise meaning to the idea that T(A)/ is 'less concentrated' than / wedefine upper and lower truncations of Ll functions (definition 3.1). The uppertruncation is used to measure the concentration of / (see eq. (3.5)), and lemma 3.2then gives a precise meaning to the statement that convolutions spread. We give a'spectral analysis' of L1 functions, which allows one to handle the difficulties thatarise in an argument when L1 functions take on either very large or very small values.

Lemmas 3.3 and 3.4 are technical lemmas required for the proof of theorem 3.5.(Hint: read the proof of theorem 3.5 before reading lemmas 3.3 and 3.4.) The basicidea of the argument is as follows. One applies the AT condition to / = XA for someA<=X, a second function /2, and some e > 0 . This gives functions A,,A2 and /satisfying ||T(A,)/—/|| s e, i = l , 2 . Since convolutions spread, choosing/2 veryconcentrated relative to the set A forces / to be very concentrated relative to / , .

This in turn forces A, to be very 'spread out' (lemma 3.3). However A, being veryspread out makes it difficult to keep the support of T(A,)/ close to A, which it mustapproximate, and simultaneously to keep T(A,)/ small on Ac. In particular, whenthe partition (A, X \ A ) moves independently under T, this becomes impossible(theorem 3.5). In order to make this argument precise, one must replace A, and /by functions AsA, and g =f with better support properties. This is done in lemma3.4 by a technical application of the spectral analysis for V functions. The conditionthat T(A)g be small on Ac then forces the support of T(X)g to be too small toapproximate A. In particular, the proof seems somewhat stronger than the statementof the theorem. It suffices that kft(B)^0 (see eqs. (3.30), (3.31), (3.41), (3.42),(3.52), (3.53)).

Let (X, ft) be a Lebesgue measure space. Then (X, ft) is isomorphic to Lebesguemeasure on [0, 1]. Let /e L\(X, ft). Then one can choose the isomorphism so thatthe (transformed) function / is monotone decreasing. The upper (resp. lower)truncation of/ is a function whose graph is identical to the graph of/ except thatthe upper left hand corner (resp. lower right hand corner) has been 'chopped off'.More precisely we have:

Definition 3.1. Let fe L\.(X, ft), a > 0. We define the upper truncation of fat a by

.//«;«: a 2 )

and the lower truncation of f at a by

The continuity of upper truncations is expressed by the conditionl l / -gN e ^ | | / [ " ] -g [ a ] | |< e , (3.4)

which follows immediately from eq. (3.2). It should be noted that eq. (3.4) doesnot hold for lower truncations.

Consider the inequality

||/-/[<"]||> H/ll -v, (3.5)where a, 77 >0. If 17 is small compared to ||/||, this inequality forces most of thecontribution to the L1 norm of / to come from that part of the graph of / lyingabove a. If in addition a is large compared to ||/||, it then forces most of/ (in thesense of L1 norm) to be supported on a set of small measure. It can therefore beused as a measure of the 'concentration' of a function. The following lemma nowgives a precise meaning to the statement that 'convolutions spread'.

LEMMA 3.2. Given (X, ft, T), Xef[+(Z), ||A|| = 1, andfe L[

+(X, ft), then

|| T(A)/- (T\)/)["]|| < | | / - / [ a l | | . (3.6)Proof. If/, I \jfj<=Ll

+(X, ft) then it follows directly from eq. (3.2) that

(lA,Z)[a](x)2lA,(/ta ])(x). (3.7)

Since | | / - / [ a ] | | = 11/11 - ||/[a]||, and ||T(A)/|| = ||/||, eq. (3.6) follows immediately. •

We now proceed to the spectral analysis of L' functions. This analysis is, in effect,a consequence of considering \fdfx in terms of horizontal slices of the graph of/Let fe L'+(X, /A). For each a > 0 we define a function Ea{f) on X by

(Ea(f))(x)fl if10 if

(3.8)

We define a measure vf on R, absolutely continuous with respect to Lebsgue measureda, by

It follows that a.e. we haver oo r co

/ ( * )= (£fl(/))(x)<fa = EiJo Jo

(/) dvAa),

(3.9)

(3.10)

where

E'a(f) = (3.11)

otherwise.

We then have

= ^ dvf E'a(f) = | dvf,

and

(3.12)

(3.13)

(3.14)

(Equality in eq. (3.14) holds only when/=0.) These equations indicate the signifi-cance of the measure vf.

LEMMA 3.3. Given (X, fi, T),n° T = /*,/, e Ll+(X, /x), | | / i | | s l ande>0, then thereexists / 2 e U ( X , M ) , | |/2| | = 1, such that for any fe V+{X, M ) , | | / | | = 1, and A, ,A2€

^i .(Z), l | A , | | s l , ||A2|| = 1 satisfying

WTMf-Mse, « = 1;2 (3.15)

we have

supAly<6e. (3.16)

Proof. The proof of this lemma consists of a long sequence of inequalities. Neverthe-less it is a completely straightforward and obvious application of the above ideas.Choose a < oo such that

(3.17)

212 A Connes and E. J. Woods

Choose / 2 e L\.(X, /LA), II/2II = 1, such that

||/2-/2fle"1]||S:l-E. (3-18)

It follows from eqs. (3.4) and (3.15) that

||/[2

a£"'] - (T(A2))/[aE"']|| < e. (3.19)

Eqs. (3.15), (3.18) and (3.19) give

l|7XA2)/-(T(A2)/)[aE~']||>l-3£. (3.20)

Eq. (3.20) and lemma 3.2 give

\\f-fi''-'i\\>l-3e. (3.21)

Let

A = {x:f(x)>ae-1}. (3.22)

Since ||/| | < 1 we have

n(A)<a~xe. (3.23)

Eq. (3.15) gives for each^eZ,

fJ T-'A

^ (A,,Ty-/,) d/x. (3.24)JT'A

Eqs. (3.21) and (3.22) give

A, , r /du>A 1 ; ( l -3e ) . (3.25)JT-'A

Eqs. (3.17) and (3.23) give

/ , d M < 2 e . (3.26)

Eqs. (3.24)-(3.26) give ( l -3e )A u <3e and hence

Since convolutions preserve L1 norms and ||/|| = 1, it follows from eq. (3.15) that|| A, || > H/ill - e . Thus if e is very small, eq. (3.16) forces the support of A, to be verylarge.

LEMMA 3.4. Given (X,/J,, T), Ac X, fe Ll+(X,/x), \xe€\(Z) and e>0 such that

l|7XA,)/-*A||<e, (3.27)

then there exist g e L+{X, /*), A e €+(Z) such that

| | /-g| |<4e5, (3.28)

l|Ai-A||<£J, (3.29)

and

(3.30)

where S is the support o/T(A)g, fc is the number of elements in the support K of\, and

(3.31)

Proof. Eq. (3.27) implies that

| Ti^JA<| (3.32)

A<

It follows from eq. (3.10) that

T(A1) = j dvXl(a)T(E'a{\x)). (3.33)

Let

p(a)=\ T(E'a(X))fdn. (3.34)JAC

Eqs. (3.32)-(3.34) give

^dVxl(a)p(a)^e, (3.35)

and hencee'>}<eK (3.36)

It follows from eqs. (3.34)-(3.36) that there exists b>0 such that

p(b)<£2 (3.37)

and

f dvki{a)Jo

(3.38)

It follows that A = (A,)[b] satisfies eq. (3.29). Since K is also the support of E'b{k),eq. (3.37) becomes

>>\ duk'1 I Vf=k~x I fJ A' je K jeK J T~'AC

= [ d

J.

(3.39)jeK J

Eqs. (3.31) and (3.39) give

(3.40)IB'

It follows that g(x) = XB(x)f(x) satisfies eq. (3.28). Eq. (3.30) is satisfied by construc-tion. •

THEOREM 3.5. Let (X, fi, T) be an AT transformation, n(X)= 1, and /x° T=ix.Then the entropy h{T) = 0.

Proof. Assume that h(T)>0. By a well-known result of Sinai there is a partition{A, Ac} of X that moves independently under T (see for example [18, p. 43]). Wecan assume that 0 < / A ( A ) < 5 . Consider the set B given by lemma 3.4 (where e, f,

and A! will be chosen below). Then (JL(B) depends only on the number k of elementsin K, and can be calculated directly from the binomial coefficients. Since the specialcase fi{A) = ^ dominates, one easily obtains the inequality

n(B)<2-\ (3.41)

Choose k0 sufficiently large that

fco2-*°<iM(A), (3.42)

and choose e > 0 sufficiently small that

e + 5e5<i/U,(A) (3.43)

and

12e*o</i(A). (3-44)

Now let /i(x) = XA(X). By lemma 3.3 there exists / 2 e L+(X, fj.), ||/2|| = 1, such thateq. (3.16) is satisfied. Since T is AT there exists fe L$X, n), ||/|| = 1 and A,, A2e<?+(Z), ||A2|| = 1, such that

\\T(X,)f-f\\^e, « = 1,2. (3.45)

By lemma 3.4 there exist A and g = Xnf satisfying eqs. (3.28)-(3.30). Eqs. (3.28)-(3.29) and (3.43) give

\\n$g-XA\\*&(A), (3-46)and hence

M(S)>^(>*), (3-47)where S is the support of T(\)g. Since ||g|| s ||/ | | = 1 and convolutions preserve Vnorms, it also follows from eq. (3.46) that

| | A | | > 1 M ( A ) . (3.48)

Since A =(A,)[()], eq. (3.16) implies that we also have

supA(j)<6e. (3.49)ieZ

Eqs. (3.48)-(3.49) give

6efc>i/x(A), (3.50)

where k is the number of elements in the support K of A. Eqs. (3.44) and (3.50) give

k>k0. (3.51)

Eqs. (3.51), (3.41) and (3.42) give

(3.52)Eq. (3.30), which was obtained from the requirement that T(At)/ is small on Ac, jnow gives j

p(S)<&(A) (3.53) j

which contradicts eq. (3.47). • i

4. Constructions of the flow of weightsIn order to make our exposition reasonably self-contained, we give here threedifferent constructions of the flow of weights, each of which will be used at some

point in our argument. We also construct some measures on the resulting spaces.For the proofs of all statements in this section, see [5].

Discrete construction. Let M be a type III0 factor acting on the Hilbert space H,with T G T ( M ) , r>0 . ( I f7~(M) = {0} one is then forced to use one of the followingtwo constructions.) Let 4> be a faithful state on M, <x* = l. Let (en)n e Z be thecanonical orthonormal basis for €2(1.), and define

Sen = en+U (4.1)

p,en = X"en, (4.2)

where neZ and A = exp (—2TT/ T). The equation wA(x) = Trace pKx defines a faithfulsemifinite normal weight wA on J£(f2(Z)). We have

o-7>(S) = pi{Sp-A'l = \"S, (4.3)

and

<ox(SAS*) = X(oA(A). (4.4)

Let

\ (4.5)

(4.6)

(4-7)

S = S®1. (4.8)

Then af = a"h®af and it follows from eq. (4.3) that the automorphism 0 = AdSleaves the centralizer N = Mj, invariant. Eqs. (4.4), (4.7) and (4.8) give

4>°6 = \4>. (4.9)

The centre C of N is isomorphic to L°°(B, vB) where (B, vB) is a Lebesgue measurespace. 0 then defines an automorphism, which we shall also denote by 6, of {B, vB).The flow of weights for M is the flow (X, v, F,) built over the base transformation(B, vB, 6) with a ceiling function of constant height 2TT/T. Furthermore

f= \

J

fN=\ N{b)dVB(b), (4.10)

J B

where the N(b) are type II^ factors. If M is injective then the N(b) are all(isomorphic to) the unique injective II,*. factor R0l (see [4]). One can then write

c©N = ROJ®LC°(B, vB)=\ ROtldpB(b), (4.11)

J B

andre

4> = T®VB=\ rbdvB{b), (4.12)

where T is a trace on JR0,I and Tb is a trace on N{b).We now assign to certain positive operators in N, measures /J. on B, fi < vB. Let

f©A= A(b)dvB(b)eN+ (4.13)

JBI B

be such that

$(A) = | rb(A(b))dvB{b)«x>. (4.14)J B

The equation

fiA(c) = $(Ac), ceC (4.15)

defines a finite measure fiA< vB. It follows from eqs. (4.14) and (4.15) that

(dnA/dpB)(b) = rb(A(b)). (4.16)

It follows from eq. (4.16) that if, e, f are ^-finite projections in N, then /xe = nf ifand only if e~f in JV. Since N(b)~ N(b)®R0J one can construct families ofprojections eb(a)e N(b), a>0 such that

rb(eb(a)) = a, (4.17)

and eb(f(b)) is a vB-measurable family of projections for any feL'+(B, pB). Itfollows that every finite measure fx<vB occurs as /u,e for some e, namely

•-rJ B

dvB. (4.18)

Continuous construction. Let M be a type III0 factor acting on the Hilbert space H,4> a faithful state on M. On L2{U) we define

(VJ){t)=f(t-s), (4.19)

and

(pf)(t) = e'f(t). (4.20)

The equation w(x) = Trace px defines a faithful semifinite normal weight o> oni?(L2(R)). We have

<(Vs) = e-ilsVs, (4.21)

and

w{VsAVf) = e'(o(A). (4.22)

Let

(4.23)

(4.24)

w = w®0, (4.25)

V^=VS®1. (4.26)

Then 6SN6;' = N where N = Mj and 0S = Ad Ws. We have

c©JV= N(x)dv(x), (4.27)

Jxwhere the JV(x) are type II^ factors, and the centre C of N is isomorphic toL°°(X, i/) where (X, ^) is a Lebesgue measure space. The automorphisms 6S definea flow Fs on (X, i-) which is the flow of weights for M If M is injective then

N(x)~R0J [4], and one can write

rei,v)=\ R^dv{

Jx= R0l®Lcc{X,v)=\ Roxdv{x) (4.28)

IX

andc©

U = T®V= rdv (4.29)

where r is a trace on /?0,i-The construction of measures in the continuous construction is quite analogous

to the discrete case, except that they occur on the flow space X rather than the basespace B. Let

- J : A(x)di>(x)eN+ (4.30)Jx

be such that

«5(A)=| Tx(A(x))dv(x)«x>. (4.31)Jx

Then the equation

HA(C) = d i ( A c ) , ceC (4.32)

defines a finite measure \xA < v such that

(dfiA/dv)(x) = Tb(A(x)). (4.33)If e, f are a5-finite projections in N, then fie = nf if and only if e ~f in N. As inthe discrete case, every finite measure fi < v occurs as fie for some projection e. Weshall need the following lemma.

LEMMA 4.1. Let (Y, a) be a measure space, and let ey be a cr-measurable family ofdi-finite positive operators in N such that

is ii-finite. Then

Proof. We have

e= eydo-(y) (4.34)J Y

fie= | fierdtr(y). (4.35)J Y

f©e= e{x)dv{x) (4.36)

Jxand

f®ey = ey dv(x). (4.37)

Jx

Eqs. (4.34), (4.36) and (4.37) imply that

e(x) = I ey(x) dcr(y) (a.e. v). (4.38)JY

Eq. (4.38) implies eq. (4.35). •

Lacunary construction. Let M be a factor of type III0, 4> a faithful lacunary weightof infinite multiplicity on M (lacunary means that 1 is an isolated point in Sp A,,,).Then

\ M(b) dvB{b), (4.39)B

where the M(b) are type Hoc factors, and the centre C^ of the centralizer M$ isisomorphic to L°°(B, vB) where (B, vB) is a Lebesgue measure space. If M is injectivethen M{b)~ROi. There exists p e Q , 0 ( UxU*) = cf>(px), xeM, (4.40)

tU}", (4.41)and

UM^U*^M^. (4.42)

Then 6 = Ad U defines an automorphism, which we also denote by 0, of (B, vB).The flow of weights for M is the flow (X, v, 9S, sent) built over (B, vB, 6) withthe ceiling function p, where

X = {(b,t):beB,l>t>p(b)} (4.43)

and &s{b, t) = (b, e"sr)if 1 > e~st> p{b) with the obvious extension to other valuesof 5.

We again construct certain measures on the flow space X. Let ifi e M*. Then thereexists h 6 M j such that s(h)p s /j < 1,1 - h is non-singular, and there exists a unitaryue M such that

.KX) = 0(/IE(IIXM*)), (4-44)

where E is the conditional expectation from M onto M^. We have

\ hbdvB{b) (4.45)B

where s{hb)p(b)< hb< 1. L e t / e L°°(X, v). Then the operator

f®*/(*)= hbfb(hb) dvB(b), (4.46)

JBwhere /fc(r) =/(b, f), is well-defined. The equation

^(f) = <f>(¥(h)) (4.47)

defines a measure /x , on the flow space X (which we will sometimes write as p,h).(Eqs. (4.47) and (4.32) are related as follows. Let 4i = u>e, eeM^. Then /ij, = /i,e.)Some terminology is helpful at this point.

Definition 4.2. Let (X, v, F,) be a flow. We call the measure fi on X smooth if p. < v,and smoothable if

/ * / * = ( dtf(t)FtlL (4.48)J —GO

is smooth for all / e LX+(U, dt).

The finite weight ip is integrable if and only if / ^ is smooth. The smoothablemeasures are precisely the measures of the form J ixbdvB(b) with respect to somebase and ceiling function construction of the flow. It is then obvious from eqs.(4.45)-(4.47) that given any smoothable measure /A, one can choose h,,, beB sothat <p(hf(h)) = fi.(f),fe L°°(X, v). I.e. every smoothable measure fi is of the form^ for some i/i e M^..

5. Comparison of finite periodic weightsLet M be a type III0 factor with TeT(M) for some 0 < T<oo, and let 0 be afaithful state on M, v%= 1. Let A, M, 0, S, M#, C+, 0, B, vB be as in eqs. (4.1)-(4.12).To each finite weight I/»G M\ (see definition 5.1) we associate a finite measure fi^,on B, fjL.^, < pB. We establish a number of properties of the map ip -* fj.^. In particularlemmas 5.7, 5.8 and 5.9 are required for the proof that AT implies ITPFI in thediscrete case T(M) ^ {0} (see lemma 8.2).

If ip is a weight on M such that a%= 1, then (Dip: D<f>)T = e"*s(ip) where 0 < a <2TT, and (Dip: D0),, l e R is the cocyle Radon-Nikodym derivative (see [5, pp. 478-479]). The weight 0 = \p®<l> will satisfy a-% = 1 if and only if a = 0. Note that forj8 > 0 we have

so that for some A < ^ < 1 we have (D(fiip): D<f>)T = s(ili).

Definition 5.1. Let a> be a faithful weight on the von Neumann algebra si such thato-r = 1 for some 0 < T < oo. Then siT

m denotes the set of all finite weights ip on sisuch that (Dip: D<o)T = s(if/). If u is a partial isometry in si with uu* e siw then theequation ip(x) = w(uxu*), xesi+ defines a weight ip with support w*w. We write

Definition 5.2. Let w and ip be weights on a von Neumann algebra si. We say thata) and ip are equivalent and write u> ~ ip if there exists a partial isometry u e si suchthat uu* = S(CJ), u*u = s(il/) and ip = wu.

LEMMA 5.3. Let M, T, <f>, M, <£ be as above. Let ip e M J. Then there exists a projectione e M$ such that ip ~ 4>e. If e, f are projections in M<f then 4>e ~ <pf if and only if e ~fin M$.

Proof. The assertion <f>e ~ <$>f if and only if e ~f in M$ is lemma 1.4(d) of [5]. Toprove the first assertion, consider the weight 0 defined on P = M®F2 by

2 \

X xij®eij) = 4>(xu) + ip(x22). (5.2)

From ([2, lemma 1.2.2]) and ([5, pp. 478-9]) we have

(5.3)

(5.4)

and

o-°(x®e22) = crt(x)®e22 (5.5)

for all x e M , where u, = {D\jf. D$)t, teU. By [5, lemma 1.4(b)]) we have ^i~4>e

for some e if and only if

s(il>)®e22<l®eu(Pe), (5.6)

(i.e. s(i^)®e22 is equivalent in the centralizer Pe to a sub-projection of l®e n ) .Since M$ is properly infinite (see eq. (4.10)), there exist projections e,e M$,jeZsuch that e}ek = 0 if/ ^ k, e, ~ ek and £ e, = 1. It follows from eq. (5.3) that e, ® e, [ € Pg,ej®en~ek®exx(Pe), and hence that l®e,i = E / 6 2

e j®en is a properly infiniteprojection in Pe. Hence to prove eq. (5.6) it suffices to show that given any projection/;= s(i//), / # 0, / e M^, there exists yePe such that

( / ® e 2 2 M l ® e u ) * 0 . (5.7)

Since aeT = 1 it follows from eq. (5.4) that u, is periodic with period T and hence

M,= I u ( t ) c i 2 * / T , (5.8)IceZ

where

* Jo^ | X)e-i2«ktlT. (5.9)* Jo

Now

(5.10)

and hence fu(K)*0 for some X e Z. Eq. (5.9) gives

o-?(«<K)®e2,) = e l 2 " K ' / V K ) ®e 2 1 . (5.11)

It follows from eq. (4.3) that the unitary S defined by eq. (4.8) satisfies

o?(S) = e-'2i r ' /7S, (5.12)since A" = e-'2irI/T. Define

>> = (u ( K )®e2 1)(S-K®e i l ) . (5.13)

Eqs. (5.11)-(5.13) give <r?(y) = y, hence ye Pe. Since S is unitary and/w(K)5^0, eq.(5.7) is satisfied. •

Definition 5.4. Let M, (#i be as above, ^ e M J. We define the measure /t , associatedwith t/> as the measure fie defined by eq. (4.15) where \ji~ ^>P

LEMMA 5.5. Let ^r,, i/»2€ M j . TTien ^ L = u. if and only ifip, ~ i/»2.

Proof. The lemma follows immediately from lemma 5.3 and the fact that ne = nf ifand only if e~f in M$ (see eq. (4.16)). •

LEMMA 5.6. Let «/»I,I/»2€ Ml be such that s(i/»i)s(t/'2) = 0. 77»e« /U.1/,I+^2 = /U.1/,, + /A<^.

Plroo/ Note that eq. (4.16) implies that if e = e1 + e2 where e, eue2eM^, thenMe= Me, + /*c2- By lemma 5.3 we have ty = <}>Uj, j =1,2 where, since M$ is properlyinfinite, we can choose u, and u2 such that exe2 = 0 where e, = M^W,. Then u = M, + M2

is a partial isometry such that i/f, +1/>2 = <£u, and M*M = et + e2. Since < u ~ <£„.„ and< UJ ~ 0e/ the result follows. •

L E M M A 5.7. Let ipeM$. Let /* , , . . . , fj.n be measures on B such that / t 0 =Zj=i Mr Then there exist orthogonal projections eu...,en such that s(tp) =X"=i ej

Proof. Since every fi<vB occurs as nP for some p e M J (see eq. (4.33)), there existmutually orthogonal projections fu... ,/„ e M and p , , . . . , pn € M j such that

*(PJ)=J5 (5-14)and

AtPj = M> j = l , . . . , « . (5.15)

By lemma 5.5 and 5.6, p = Z P/ ~ </>• Hence there is a partial isometry u withM*M = S(P) = £./;, uu* = s(if/), and <A = pu. Then e, = u/J are the desired pro-jections. •

LEMMA 5.8. Let t/f 6 M J, </> ~ <£c. T7ien

^M* = Ap-A-V = AM»(e)-

Proof. We have ^ = ju.e. Let ce C,j. Using eqs. (4.9) and (4.15) we have

= ($ o »)(«(e)c) = A<£(0(e)c) = A/t.(«)(c). (5.16)

It remains only to prove that /AA* = Me-'<*)- Let u be a partial isometry in M suchthat <j) = 4>u and uu* = e. Then

(A«A)(X) = A<MUXU*) = <M0-|(MXW*)), xeM. (5.17)

Since fl = AdSwe get

(A<A)(x) = <^(S«xu*S*) = 0 & (x) . (5.18)

But £ , - 4 ^ . and SMU*S* = »-'(e). •

LEMMA 5.9. Let I/J, e M J, i/», # 0, and e > 0. Lef jt2 5 0 fee a inife measure on B,fi2< vB. Then there exists </>2e M j such that

(i) J ( * 2 ) = S ( * , ) ;

(ii) M*2 = M2; a«d(iii) H^-^H^IlM^-^ll + e.

Proof. Write (1, = ^ and fj = dfij/dpB, j" = 1,2. By a routine argument one canchoose a family of projections e(a)e ROi, 0<a<oo such that

e(a)e(P) = e(a) ifa</3 (5.19)

and

r ( e ( a ) ) » a (5.20)

where T is the trace on ?0,i such that <£ = T ® J/B (see eqs. (4.11) and (4.12)). ThenHj = Hej,j=l,2 where

e(fj(b)) dvB{b). (5.21)

Let

B+ = {b:f2{b)>Mb)}, (5.22)

(5.23)

(5.24)

Then

wheree, =

reeJ+=\ e(fj

JB+

dvB(b)

(5.25)

(5.26)

etc. Let M be a partial isometry in M such that (/>, = <£„ where MM* = e,, u*u = 5(i/>,).

Case (i). vB(B+)= vB(B-) = 0. Take ^2 = -

Case (ii). vB(B+), vB(B_)>0. Let w be a partial isometry in M mapping thenon-zero projection e2+ - e1 + onto the non-zero projection e,_ - e2— Define i//2 = </»„where

ll> = $el-(ei_-e2_)+(te2+-ej»- (5.27)

Then ifi~ <t>€2 so that fi^, = fi2. Furthermore s(i/») = e,, s(tp2) = M*M = s ^ i ) , and

1 1 * 2 - ^ 1 1 = 1 1 ^ - ^ , 1 1 . (5 -28 )We have

* - &, = (^^+-e,+)« - ^, ,- e 2_- (5-29)

Since \\4>f\\ = $(/) = fif(B) we obtain

Case (iii). vB{B+) = O,

and

Let

0. Choose g(fo), fceB such that

0< | g{b)dvB(b)<ke.J B

g= I e{g{b)) dvB(b).JB

(5.30)

(5.31)

(5.32)

(5.33)

Let w be a partial isometry in M such that a>*a> = g, oxo* = e{ — e2 + g. Define </>2 = i/»u

where

xjj = 4>e2_g + {4>g)o>. (5.34)

Then ip ~ 0<,2 so that / ^ = fi2. Furthermore s(i/f) = e, so that s(tp2) = u*u = s(ip{), and

11^-^11 = 11^-^,11. (5-35)Since

& , - * = <£, + <£«,-«,-(&). (5-36)and

- J= | g{b)dvB<\e

we obtain || i/>2 — B(B+)^0, ^B(B_) = 0. The argument is similar to case (iii).

(5.37)

•

6. Comparison of finite weights: the general caseWe extend the results of § 5 to the general case. This is straightforward except forlemma 5.9, where we now use the lacunary construction (see lemma 6.4). In thissection M is a type III0 factor, and we follow the notation of § 4.

LEMMA6.1. LetipeM^., thenixx^, = X^-'/J,^. Ifif/U t/»2e M% tfien t/f,

and M^,+^2 = M , + M*2 if s(<l>i)s(4>2) = 0-

Proof: This is corollary 1.13 (ii) of [5]. •

LEMMA 6.2. Let tpeM%, /x^, = Y,"= i /*;• Then there exist orthogonal projectionseu...,en such that s(<l>) = I " = I e, and </» = I J = 1 tk, where /u,^. = MrProof Since every smoothable measure occurs as /J.X for some \ (see eq. (4.48) etseq.) the proof of lemma 5.7 holds verbatim. •

The next lemma is a technical result needed in the proof of lemma 6.4.

LEMMA 6.3. Let o-x, cr2 be non-atomic measures on I = [0, lj.cr^/) = cr2(I) < °o. LetFj(x) = o-j([0, x]), and let

Sj = {xel: Fj(y) = Fj(x) implies y = x},

7" = 1,2. Then cr,(/\S,) = 0 and the equation F,(y(x)) = F2(x) defines a monotonicbijection y: S, -» S2 satisfying

f{y(t))do-x{t)=[ f{t)da2{t)Jo

for allfe £,'(/, a2). Furthermore

\t-y{t)\do-M. (6.1)fJo

Proof. To prove eq. (6.1) consider the function / defined by

f + 1 \U>y(t)

f'U) = \ 0 i f f = y (O , (6.2)

(-1 ift<y(t),

and/(0) = 0. Since H/IU^ 1 we have

=l\t-y(t)\do-M. (6.3)

All other properties are obvious. •

LEMMA 6.4. Let i/»i be a finite integrable weight on M, and let /JL2 be a smooth measureon X (i.e. /x2< v, see definition 4.2).Then there exists i/»2e M^ such that:

(i) s(*2) = «(*,);(ii) fit2 = n2; and(iii) ||i/',-</'2||

Proof. We use the lacunary state construction of the flow of weights (eqs. (4.39)-(4.48)) where we choose the lacunary state 4> so that j »>= dvB{b)\ tdaub(t), (6.5)JB J p(b)

where we can choose the measures <rub so that fo r / e L\X, MI) we have

M.(/)=f dvB(b) P f/(M)<K>(0 (6.6)JB Jp(b)

where Mi = M*,« By the remark following definition 4.2, we can choose measures o-2,bsuch that

[ tf(b,t)d*2tb(t). (6.7)B J p(b)

We begin by altering the measures slightly so that lemma 6.3 can be used. Let

B+ = {beB:<ru([p(6),l])><r2,t([p(()),l])}, (6.8)and

B_ = B\B+. (6.9)

For be B+ define t2tb = 1 and

U,b= sup{p(b)^t^\:aub([p(b),t]) = o-2>b([p{b), I])}, (6.10)

and for b e B_ define tlb=l and

t2.6 = sup{p(6)srs.l:a'2>fc([p(6),0) = or1,t([p(fc),l])}. (6.11)

Define /*;, / 4 by

( d M V ^ ) ( ^ O = A'[P(b,.(,h](O. (6.12)Then

II M i " Mi 11 = P>i (*"[!, .„!])

(6.13)

where the last inequality follows from the fact that dfij/duB ° o-j=t>\. Similarly

| | M 2 - M 2 N 2 | | M , - M 2 | | . (6.14)

We can now write MJ = VB ° c'j, j = 1,2 where

<b([p(b),l]) = <7'2,b([p(b),l]), beB. (6.15)

By lemma 6.3 there exists a measurable function y(b, t) such that for a l l /e L'(X, p.\)we have

I y(6,0/(6; y(fc, 0) <K»(0 = | »/(*, 0 Ari.*(0, (6.16)Jpfb) Jp(fc)

for a.e. b e B. We definef© f'e

h\= dvB{b) tda'ub(t) (6.17)JB J p(b)

h'2= I dvB(b) I * y(b, t) da\,b{t), (6.18)JB Jp((>)

and

t/0'(x) = <£(/.,'£(«*«*)), 7 = 1 , 2 . (6.19)

Using eq. (6.13) we obtain

| | ^ , - ^ | | = *(|*,-fcl,|) = ||/*1-/*ill=s2||AiI-i»2||. (6.20)

We also have

'ub-alb\\ = \\n\-,j,'2\\, (6.21)

where the last inequality follows from lemma 6.3.

Case (i). fi\ = fix and /JL'2 = (JL2. Then i//2 = * 2 satisfies the lemma.

Case (ii). nt* fi\ and n2* (JL'2. Let h"x=hx-h\. Then s(h'{)s{h\) = 0 and s(li5')^0.Construct /i2' such that s(A2) = s(AJ) and /*fc2" = fi2-fi'2. Let h2 = h'2 + h2 and define

tli2=(t>(h2E(uxu*)). (6.22)

By construction ^ = /A2 and 5(^2) = •s('Z'i)- We have i/>2 = *l>'2+ <l>2 where

11*511 = 0(|*2|)=ll/*2-M2||=s2||/*i-/*2|| (6-23)

(see eq. (6.14)). Eqs. (6.20), (6.21) and (6.23) give

| | * , - * 2 | | = S 5 | | A I 1 - / I 2 | | . (6.24)

Case (iii). fi, =/tJ and /LI2^M2- Choose a projection e e M such that 0 < e < s ( / i 2 ) ,and

0(efcj)=s HM1-M2II. (6-25)

Choose h2' such that s(ft2') = e and

M/.5=M2-M2 + Mefc2- (6.26)

Then eqs. (6.14) and (6.25) give

| | / * M | | S 3 | | M I - M 2 | | . (6-27)

Define

4,2(x) = <j>((h'± + (l-e)h'2)E(uxu*)). (6.28)

By construction ^i^ = ix2 and s(t//2) = s(*l/i). Eqs. (6.25) and (6.27) give

ll*2-*i|| ^411/4,-/4211. (6-29)Since ip, = ij/'u eq. (6.21) now gives

| |*,-*2| |=s5| | /*1-/*2 | | . (6.30)

Case (iv). /u., ?* fi\ and /t2 = p'2- Choose a projection ee M such that 0 < e<s(ft'i) =s(h'2) and

Mi-A*2ll- (6.31)

Choose h'i such that s(h$) = s(hl)-s(h[) +e and

f*hi = Hehi- (6-32)

Define ip2 by eq. (6.28). By construction /tfc = /A2 and s(<//2) = 5(i/»,). Eqs. (6.31),(6.32) and (6.28) give

llfc-^II^H/u.-^ll. (6.33)Eqs. (6.20) and (6.21) now give

| | ^ - * 2 | | £ 5 | | M l - / t 2 | | . (6.34)

D

7. Product property and ITPFI factorsWe introduce the 'product property' (definition 7.1) which is a variation of St0rmer'sproperty of being 'asymptotically a product state' [19]. This is a technical propertywhich is equivalent to ITPFI (corollary 7.4 and lemma 7.6). Its purpose is to simplifythe task of verifying the ITPFI property by eliminating the iterative part of theargument.

Definition 7.1. Let M be a von Neumann algebra. A finite weight <f> on M is saidto have the product property if given e > 0, a strong neighbourhood V of 0, andx,,... ,xne M, there exists a finite type I factor K<^ M and finite weights <f>u <f>2

on K, Kc = K'nM respectively such that:(i) Xj; e K + V, j = 1 , . . . , n, and

(ii) ||^-<h®«fcll<e.If M has a faithful finite weight with the product property, then M is said to havethe product property.

Remark 7.2. It follows immediately from the above definition that the productproperty implies approximately type I (see [8]). Clearly one can require that $, and<f>2 are faithful, and (ii) can be replaced by

If one were to study von Neumann algebras of the form (§)„ {Mv, #„) where the siv

are finite-dimensional matrix algebras, the appropriate product property would beto require only that K be a finite-dimensional subalgebra.

In order to prove that an ITPFI factor has the product property we will use thefollowing martingale condition, which was introduced by Araki and Woods([1, lemma 6.10]).

LEMMA 7.3. Let M = ® " = , (Mk, <pk) be an ITPFI factor. For each neJf there is aconditional expectation En:M^ M( n ) = ®^ = 1 Mk such that

(i) En(x) -» x strongly for all x e M;(ii) En(Mlt,) = M^ where ^ ( g C . <t>k and<t>in) = ®"k=l 4>k\ and(iii) EnEm = Enifn<m.

Proof. Define En by the equation

(En(x)a,P) = (x(a®( 0 dOY/W 0 **)) (7.1)

for all xe M, a, )3 e(g)£ = 1 Hk where Mk acts on Hk and ^(.y) = ( y ^ , <&&) for allj e Mk. That £„ is a conditional expectation and properties (i) and (iii) followdirectly from eq. (7.1) (compare [1, lemma 6.10]). Condition (ii) follows from

af"\En{x)) = En(o-f(x)) (7.2)

which in turn follows from the observation that

A4>=A^®A^kZn+lM (7.3)

and a routine calculation using eq. (7.1). •

COROLLARY 7.4. Any ITPFI factor has the product property.

Proof. Lemma 7.3 implies that any product state 0 <$>k has the product property. •

LEMMA 7.5. Let M be a von Neumann algebra with the product property. Then anyfinite weight \\i has the product property.

Proof. Let e > 0, V a strong neighbourhood of 0, and xu... ,xne M. Let <j> be afaithful finite weight on M with the product property. It follows from the Hahn-Banach theorem that the set of all finite weights x s u ch that x £ A</> for some A > 0,is norm-dense in M^. It then follows from [17, theorem 1.24.3, p. 76] that thereexists h e M+ such that

\4,(a)-ct>(hah)\<e\\a\\, (7.4)

for all ae M. By assumption there exist a finite type I factor K <= M, ke K, andfinite weights </>,, <t>2 such that

Xj<=K+V, j=\,...,n, (7.5)22\\h\\-2U\\-\ (7.6)

\^ s\\h\\-2, (7.7)

and

||fc||<||fr||. (7.8)

That one can achieve eq. (7.8) follows, as in the proof of the Kaplansky densitytheorem, by approximating an element h'e M such that h =2h'(l +(h')2)"' (see [7,p. 44]). Since

4>{hah) = 4>(kak) + <t>((h-k)ah) + 4>(ka{h-k)),

it follows from the Cauchy-Schwarz inequality and eq. (7.6) that

\ct>{hah)-4>(kak)\<2e\\a\\. (7.9)

Eqs. (7.4), (7.7) and (7.9) give

l k - 0 i ® « M ^ 4 e , (7-10)

where <p\{a) = <t>^kak), ae K, and ^2(«) = </>2(«), ae Kc. •

LEMMA 7.6. Let M be a properly infinite von Neumann algebra with the productproperty. Then M is ITPFI.

Proof. Let i/< be a faithful state on M. Let (Xj)j£N be dense in the unit ball M, ofM, let Vj be a sequence of strong neighbourhoods of 0 decreasing to 0, and let

7}n > 0 with

I ifc < 1 . (7.11)

We will construct a sequence of mutually commuting finite type I factors K{ andfaithful states <£, on Kt such that

xJeK(n)+Vm j=l,...,n (7.12)

where

)1, (7.13)\ i = 1 /

and

where (K{n))c = (K{n))'n M. It will then follow that M ~0nEN{Km<f>n).By assumption there is a finite type I factor X, and a faithful state $, such that

eqs. (7.12) and (7.14) are satisfied for « = 1 (use remark 7.2). Now assume thatKu...,Kn and <j>{, ...,</>„ have been chosen so that eqs. (7.12) and (7.14) aresatisfied. Let e\"\ i,j = 1 , . . . , kn be a complete set of matrix units for Kin), and lete = e\"K Since M is properly infinite, e ~ 1 and Me = eMe ~ M. In particular Me hasthe product property. Let W b e a strong neighbourhood of 0 such that the sum ofany k2

n elements from W must lie in Vn+1. There exists a finite type I subfactor Ln+1

of Me, yi"lj e !„+,, i,j = 1 , . . . , kn, k = 1 , . . . , n + 1, and a state <f>n+l such that

en Vî/Ci"' - e{£)xkei£) eW, (7.15)

and eq. (7.14) is satisfied with n replaced by n + 1 (where we have used the canonicalidentification of Me with K(n)c). Let

kn

y(kn) = I îfyîjeê K(n+'\ (7.16)

fc=l,...,n + l, where Xn + I is the finite type I subfactor of M obtained from thecanonical embedding of Ln+l. Then

y kn ) - j c k €v B + l

so that eq. (7.12) is satisfied. Now let <f> = ® " = 1 <t>k- It follows from eqs. (7.11) and(7.14) that \\4>~ »ft|i < I- Hence the representation TT , of the UHF C*-algebrasi = UjePsj Kj induced by <f> is unitarily equivalent to TT , (see for example [16, theorem2.7]). Eq. (7.12) implies that TT<,{S4)"=M.

Remark 1.1. By further arguments, which we omit, one can show that in the generalcase the product property implies ITPFI.

8. ITPFI factors and AT flowsIn this section we prove our major result, namely the equivalence of the ITPFI andAT properties (theorem 8.3). The proof that ITPFI implies AT (lemma 8.1) is ratherstraightforward. It follows from the known existence of a factor martingale (lemma

7.3), together with the relationship between certain positive operators in the cen-tralizer and finite measures on the flow space (see eqs. (4.13)-(4.18) and (4.30)-(4.33)). The converse is obtained by proving that AT of the flow of weights impliesthe product property (lemma 8.2).

LEMMA 8.1. Let M be an ITPFI factor of type III0. Then the flow of weights for Mis AT.

Proof. Since the proof for the discrete case T(M)^{0} is much more transparentand, furthermore, motivates the proof for the general case, we present it first.

Case (i). T(M) * {0}. Let Te T(M), T*0. Using either lemma 11.2 of [1] or resultsfrom [2] we can write M = ® ^ = 1 (Mk, (f>k) where ap = 1 for all k. We will use thediscrete construction of the flow of weights here. Let A, M, <£, 6, B and vB be as ineqs. (4.1)-(4.12). We will prove that the base transformation 6 is AT. By lemma 2.5this is equivalent to the flow being AT.

Let e > 0, and let fi,,..., ixn be finite measures on the base space B, fit,..., fin<vB. Using eqs. (4.15) and (4.18) we obtain projections eêM^ such that ixei = fj.j,j = 1 , . . . , n. It follows from our lemma 7.3 and lemma 2.3 of [14] that there existw<oo and positive operators J$ e M(J.J> where Mim) = ®k=0 Mk, </><m) = o>A®( ® r . i <h) and M0 = i?(/2(Z)), such that

4(\ej-Ji\)ê, j=l,...,n. (8.1)

Since e,, f e M^ we have the inequality

\*((ej-fj)c)\*\\c\\t(\ej-/j\), ceM,

(see for example, [7, lemma 11, p. 63]. It now follows from eqs. (8.1) and (4.15) that

\\ne.-/jLfl\\<e, j=l,...,n. (8.2)

The proof will be completed by showing that

J 5 = I I cjklejkl, 7 = 1 n, ( 8 . 3 )keZ 1=1

where cjU > 0, eJk, are minimal projections in M( ?2>, ejU ~ 0ke where e is a fixedminimal projection in M$l>, and N<oo, Eqs. (8.3) and (4.15) and lemma 5.8 thengive

%, = I cJkk ~kekne, j = 1, . . . ,« , (8.4)

where cjk=Ydl cjkl. The AT now follows directly from eqs. (8.2) and (8.4). Eq. (8.3)will follow directly from the structure of M7™> which we now determine.

Recall that if \p{x) = Trace(px), x e P i s a weight on a type I factor P and pe P,then af = Ad p" and hence P^, = {p}'. Thus the determination of the structure ofMffii becomes an elementary exercise in linear algebra. We have <j>(m)(x) =Trace(p(m>x), x e M ( m ) where p ( m ) e M ( m ) has eigenvalues rik=oA*,;(*> w h e r e

{^kj}j=i,...,nk is the eigenvalue list of (Mk, <j>k) and nfc<oo for fcÔ. Since o-p = 1,all ratios Afcr/Aks are precisely some integral power of A. For k = 0, each Ap, peZoccurs precisely once as an eigenvalue. Hence the eigenvalues of p<m) are of the

form a\s where a is fixed, and each s e Z occurs precisely

N = fl nt (8.5)k = \

times. It follows that

M^ = {pim)}'=®keZM(k), (8.6)

where M(k) = FN, and eqs. (4.1), (4.2) and (4.8) and 6 = Ad S imply that 6(M(k)) =M(k+ 1). Eq. (8.3) now results directly from diagonalizing f.

Case (ii) is the general case. We use here the continuous construction of the flowof weights. Let M, a>, ds, X and v be as in eqs. (4.19)-(4.29). Let e>0 , and letfti,..., fin be finite measures on the flow space X, fit,..., fin<v. Precisely as incase (i) there exist ejeM^, such that fjLe. = (ij (see eqs. (4.32), (4.33)), and positiveoperators fj e M^lt for some m <oo, satisfying eq. (8.2). The same argument usedabove shows that this time we have

y dtM(t), (8.7)

where M(t) = FN and TV is again given by eq. (8.5). Let fe M ^ L Then

and

where

where

and

/=J dtf(t),

f©ej=\ dtf(t-s),

^(y)= dte'rifit)),

T is the trace on FN. Diagonalizing the positive operators f,

J5/ = J ° dtMt),

(8.8)

(8.9)

(8.10)

we obtain

(8.11)

(8.12)

(8.13)

where Cj,(t) > 0 and efl{t) are minimal projections in M(t). Since &K./J) = fi/peq. (8.10) implies that e'c,,(r)e L'(K). Since the transitive action of U on U is AT,there exist A,,£L'(IR) and c(0, e'c(r)e Ll(U) such that

r< E N " ' . (8.14)cj,(t)- | dsAj7(s)es+'c(s+0

Let

- J dtcj,(t)e(t), (8.15)

where e{t) is a measurable family of minimal projections in M(t). Since e(t) ~ ej,(t)in M(t) we have Fjl~fjl in M(

07i> and hence

Let

f®C= dtc{t)e(t), (8.17)

and

i dtgjl(t)e{t) (8.18)

where

&,(0 = j d s A ^ e ' ^ s + O. (8-19)

Then

Gj,= ds\j,(s)esesC. (8.20)

Lemma 4.1 now gives

MG7 = J ds kj,(s)e%fic. (8-21)

Using eqs. (8.10), (8.14), (8.15), (8.18) and (8.19) we obtain

tel\cl{t)-gj,(t)\^eN-1. (8.22)

The AT of 0, now follows from eqs. (8.2), (8.11), (8.16), (8.21) and (8.22). •

LEMMA 8.2. Let M be a Krieger factor of type III0. If the flow of weights is AT, thenM has the product property.

Proof. Let e > 0, x , , . . . , xp e M, V a strong neighbourhood of 0 in M, and 0 afaithful state on M. We give first an outline of the argument. Connes' martingalecondition [3] gives the existence of a conditional expectation E onto a finite-dimensional subalgebra TV such that <j>° E = 4>, and xu ... ,xpe N+ V. If TV werea finite type I subfactor, the condition <f> ° E = <f> would imply that </> was a productstate relative to M = N® Nc and the product property would be trivially satisfied.The strategy is to embed TV in a finite type I subfactor P so that </> is approximatelya product state. The basic idea is to use the AT condition to construct a new state4> such that \\<j> - </>|| < e, and a set of matrix units for P which are eigenvectors oferf (which implies that \\i is a product state relative to M = P®PC). More precisely,one selects a minimal projection e(fc) from each full matrix algebra in TV. A measurefik is associated with each e<k). Using the AT condition the fj,k can be approximatedby measures fi 'k = X Mfc/- "The /xki determine both the desired subprojections of thee(k) (which are minimal projections in P), and the state tj/.

We give first the argument for the discrete case 7 (M)^{0} . The argument forthe general case then proceeds in exactly the same way.

Case (i). There exists T e T{M), T * 0. Let <j> be a faithful state on M such that a% = 1.

Step (i). By [3] there is a conditional expectation £:M-»/V where TV is a finite-dimensional subalgebra of M such that

<£(£(*)) = </>(*) fo ra l lxeM, (8.23)

and

E(Xj)eV + Xj, j=\,...,p. (8.24)

We have

N=@Nh (8.25)

where the Nk are type Imi factors. Let <j>k denote the restriction of <f> to Nk. Then

4>k{x) = Trace pkx, xeNk, (8.26)

where pk e iVfc. Choose matrix units ek for 7Vk such that pk is diagonal, i.e.

Pke$j = XueqSij, i,j=\,...,mk (8.27)

and

Af c ,&Af c 2>-•->Ak m t>0. (8.28)

Then, since a?°E(E(x)) = o-f(£(x)), we have

o-f ( 4 ) = o-f l(«S) = (Aw/Ay)"^. (8.29)

5<e/7 (ii). In order to use the AT condition, we construct the flow of weights as in§ 4, eqs. (4.1)-(4.18). Let Po denote the projection onto the basis vector e0 of /2(Z).Let

f = P0®ek, k=\,...,n (8.30)

where ek = eku. Since Poe g(€2{T))Mk and ek e M^ (see eq. (8.29)) we h a v e / 6 M*.

Since <^(/fc) = <j>(ek)<<x>, eq. (4.15) defines measures

M* = MA fc=l,...,n, (8.31)

on the base space fi such that /xk < vB. Using the AT condition and a variant ofremark 2.2(iii) we obtain a measure n<vB and integers nk(l), k=l,...,n,l = -L,..., L such that

l lMfc-MlMn"'"!* '*, fc=l,..,n (8.32)

where the measures

Mi= +I nt(/)A-'flV (8.33)

are non-zero.

Step (iii). We now modify the state </>. We first show that 4> is determined by itsrestrictions to ME*, k= 1 , . . . , n where, in effect, it is the conditional expectation.For x e M we have

E(x)=lxkek, (8.34)

where the numbers Xy are determined by the equation

£(e*x4) = 4 4 (8.35)Eqs. (8.23), (8.26), (8.27) and (8.34) give

<£(*)= I Afc,4, (8.36)

Since

Heuxeti) = (A(eit,£(x)ef1) = A k , 4 , (8.37)

we get

i Uf,). (8.38)k=l i = i

The desired alteration of <f> will now be obtained by changing it on Mek and using

eq. (8.38) to define the new state ip on M. From eq. (8.32) and lemma 5.9 we obtainstates \fik such that

s(k)=f\ (8.39)

/•".** = /**, ( 8 . 4 0 )

and

\\4>k-4>k\\^n~]tnkle, (8.41)

where 4>k is defined on Mfk by

4>k(P0®x) = <{>k(x). (8.42)

We define i//k on Mek by

tykix)= il>k(Po®x)- (8.43)

We define t/» on M by

Eqs. (8.28) and (8.41)-(8.44) give|| <A - <£ || s e. (8.45)

Step (iv). We now embed N in a type I factor P c M s o that i/> is a product staterelative to M = P® Pc. Using eqs. (8.33), (8.40) and lemma 5.7 we obtain non-zeroprojections jj,j=l,...,qk, k = 1 , . . . , n such that

J* = I /?, (8.46)j = i

4*k= 1L tykj, (8.47)

where

kj = (<Pk)fki, (8.48)

and integers s'kj such that

V-ikl = A ^ 0 V = \-s"'es"'fiiu, (8.49)

where Sy = S y - s , , . It now follows from lemmas 5.5 and 5.8 that

<A*>~AI("i?,|. (8.50)

Since Po is one-dimensional, we can define projections e) e M by

fi 1 (8.51)and it then follows from eqs. (8.44), (8.47), (8.48), (8.50), (8.51) and lemma 1.2.3(b)of [2] that there exist partial isometries MJV e M such that

(«#)*«# = «!, (8.52)

u?Ml)* = e1, (8.53)and

*t(uX) = \ia*>u$, (8.54)

j = 1 , . . . , qk, k = 1 , . . . , n. We extend the range of the index j to 1 , . . . , mkqk bydefining

»)i = eW,l, (8-55)where j = (l-l)qk + t, 1 < t < qk, 1= 1 , . . . , mk. Finally we obtain a complete set ofmatrix units by defining

«"=«n(«J!)* (8.56)

where i = 1 , . . . , qk, j = 1 , . . . , qh k, I = 1 , . . . , n. We can now define P to be the typeI factor generated by the u". Eqs. (8.44) and (8.54)-(8.56) imply that the u" areeigenvectors of af. Hence a-f(P) = P which implies that ip is a product state relativeto M = P®PC.

Case (ii) is the general case. The argument is virtually identical. We use here thelacunary construction of the flow of weights. For this purpose we take (/> to be afaithful lacunary integrable weight of infinite multiplicity. Basically the only changeis that lemmas 5.5, 5.7, 5.8 and 5.9 are replaced by lemmas 6.1, 6.2 and 6.4.

Step (i) is identical. In step (ii) we now construct the flow of weights as in § 4,eqs. (4.39)-(4.48). In particular eq. (4.47) defines measures

Mt = M#.k, k=l,...,n, (8.57)

on the flow space X, fik < v. Using the AT condition and again a variation on remark2.2(iii) we obtain a measure p. < v, t, e U%, nk(l)eZ, I = 1 , . . . , L such that

\\Hk-V-'k\\^n-lmile, k=\,....,n, (8.58)

where the measures

/**=! ^ ( O r ' ^ / i (8.59)7 = 1

are non-zero.Step (iii) is almost identical. We work directly on M and use lemma 6.4 to obtain

states 4>k on Mc* satisfying

s{4>k) = e\ (8.60)

M*t = /*!k, ( 8 - 6 1 )

and

e. (8-62)

ip is again defined by eq. (8.44) and satisfies eq. (8.45). In step (iv) we begin bynoting that the states ifk defined by eq. (8.60) are integrable since the measures /u.'kare smooth (see eq. (4.48) et seq.). Using eqs. (8.59), (8.61) and lemma 6.2 we obtainnon-zero projections ej,j=l,...,qk, k = 1 , . . . , n such that

ek = I e) (8.63)

and

= I </%, (8.64)

where ipkj= (tpk)e1;, and t'kjeU such that

where tkj = t'kj-tu. It now follows from lemma 6.1 that

^ y - e - V n - (8-66)As before, it follows from eqs. (8.44), (8.66) and lemma 1.2.3(b) of [2] that thereexist partial isometries u£' satisfying eqs. (8.52)-(8.56), and the type I factor Pgenerated by the Uy' has the desired properties. •

THEOREM 8.3. Let M be a type III0 injective factor. Then the following are equivalent.(i) M is ITPFI.(ii) The flow of weights for M is approximately transitive.(iii) M has the product property.

Proof. By [4] M is a Krieger factor. We have the following implications: (i)=>(iii),corollary 7.4; (iii)=>(i), lemma 7.6; (i)=>(ii), lemma 8.1; and (ii)=»(iii), lemma 8.2.

•Acknowledgement. Part of this work was carried out during the second namedauthor's visit to the IHES, Bures-sur-Yvette and the University of Warwick. Hewould like to thank the Universite de Paris VI, the Institut des Hautes EtudesScientifiques, and the University of Warwick for their kind hospitality and financialsupport.

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133-252.[3] A. Connes. On hyperfinite factors of type III and Krieger's factors. / Fund. Anal. 18 (1975), 318-327.[4] A. Connes. Classification of injective factors. Ann. of Math. 104 (1976), 73-115.[5] A. Connes & M. Takesaki. The flow of weights on factors of type III. Tohoku Math. J. 29 (1977),

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[13] W. Krieger. On ergodic flows and the isomorphism of factors. Math. Ann. 223 (1976), 19-70.[14] C. Lance. Martingale convergence in von Neumann algebras. Math. Proc. Camb. Phil. Soc. 84 (1978),

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Approximately transitive flows and ITPFI factors

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